The modified Euler method starts with an Euler step, giving a
provisional value for at the next time :

The step actually taken looks like an Euler step, but with replaced by the average of at the starting point of the step and at the provisional point:

This strategy has the effect of building in the curvature term in the Taylor series expansion of the solution (5), leaving an error of per step, so a net error of in the approximation to - hence a second order method.

Carleton DeTar 2008-12-01